Energy Dependence of the Effective Debye Temperature Obtained from Low-Energy-Electron-Diffraction-Intensity Measurements

A discussion of the effect of lattice vibrations on the effective electron-ion-core elastic scattering vertex is given using the model of Duke and Laramore. The renormalization introduced by the lattice vibrations can substantially increase the number of partial waves necessary to describe this scattering. The vertex approximation using the $s$-wave part of the phonon renormalization factor and phase shifts from a realistic potential to describe the electron-rigid-ion scattering is compared with the vertex approximation using the full phonon renormalization factor and the constant phase shift $s$-wave model to describe the electron-rigid-ion scattering. Model calculations of low-energy-electron-diffraction (LEED) intensity profiles are presented for a system having the geometrical parameters of Al(100), and effective Debye temperatures are obtained for the Bragg peaks in the intensity profiles. The dependence of these effective Debye temperatures on the inelastic-collision mean free path and on the characteristic falloff of the vibrational amplitude of the ion cores with distance from the surface is investigated. Even for a constant mean free path the ${{\Theta }_D}^{\mathrm{eff}}$ exhibit a pronounced energy dependence. By comparing the calculated ${{\Theta }_D}^{\mathrm{eff}}$ with the experimental measurements of Quinto et al., a crude estimate of the inelastic-collision mean free path in the surface region of aluminum is obtained.